To explore the spatiotemporal transmission dynamics of syphilis in Xinjiang Province, we establish a multi-stage infection syphilis reaction-diffusion model that incorporates demographic and geographic data. The well-posedness of the system is studied. Then, we derive the basic reproduction number R0, which serves as the critical threshold for disease outbreak or extinction. We also prove the global dynamics of the homogeneous model parameters and heterogeneous diffusion coefficients case by constructing the feasible Lyapunov functional. Finally, based on data from Xinjiang Province from 2004 to 2018, we estimate model parameters via the Markov Chain Monte Carlo method. The effects of treatment and intervention measures on syphilis spread are explored from both one-dimensional and two-dimensional perspectives. One-dimensional numerical simulations validate the theoretical results and show that increasing treatment coverage and strengthening public health interventions reduce syphilis infections. A two-dimensional simulation of syphilis transmission is conducted using COMSOL Multiphysics software and the finite element method, incorporating Xinjiang Province’s population and geographic data to capture regional differences. Our results suggest: (1) Syphilis infection is more severe in the southern region of Xinjiang Province than in the northern region. (2) Syphilis treatment measures in Xinjiang Province should be tailored to local conditions. Specifically, syphilis prevention efforts should focus on enhancing local healthcare and public awareness, while addressing regional disparities through tailored control strategies in Xinjiang Province.

Based on the fact that HIV/AIDS manifests different transmission characteristics and pathogenesis in different age groups, and the proportions of youth and elderly HIV infected cases in total are increasing in China, we classify the whole population into three age groups, youth (15-24), adult (25-49), and elderly ($ \geqslant $50), and establish a three-age-class HIV/AIDS epidemic model to investigate the transmission dynamics of HIV/AIDS in China. We derive the explicit expression for the basic reproduction number via the next generation matrix approach. Qualitative analysis of the model including the local, global behavior and permanence is carried out. In particular, numerical simulations are presented to reinforce these analytical results and demonstrate HIV epidemiological discrepancy among different age groups. We also formulate an optimal control problem and solve it using Pontryagin's Maximum Principle and an efficient iterative numerical methods. Our numerical results of optimal controls for the elderly group indicate that increasing the condom use and decreasing the rate of the formerly HIV infected persons converted to AIDS patients are important measures to control HIV/AIDS epidemic among elderly population.